Pell's Equation

ISBN-13: 9780387955292 / Angielski / Twarda / 2003 / 228 str.

Edward Barbeau

Opis

Pell's equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. The history of this equation is long and circuitous, and involved a number of different approaches before a definitive theory was found. There were partial patterns and quite effective methods of finding solutions, but a complete theory did not emerge until the end of the eighteenth century. The topic is motivated and developed through sections of exercises which allow the student to recreate known theory and provide a focus for their algebraic practice. There are also several explorations that encourage the reader to embark on their own research. Some of these are numerical and often require the use of a calculator or computer. Others introduce relevant theory that can be followed up on elsewhere, or suggest problems that the reader may wish to pursue. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject. Edward Barbeau is Professor of Mathematics at the University of Toronto. He has published a number of books directed to students of mathematics and their teachers, including Polynomials (Springer 1989), Power Play (MAA 1997), Fallacies, Flaws and Flimflam (MAA 1999) and After Math (Wall & Emerson, Toronto 1995).

Recenzje

From the reviews:

"This book belongs to the collection 'Problem Books in Mathematics'. Because of this choice, this book is not a course on Pell's equation but a series of exercises which presents the theory of this equation ... . This book is certainly a good book for students who are courageous enough to try to solve many exercises; they can learn a lot this way. Moreover, this long collection of exercises presents a lot of examples and a great variety of methods." (Maurice Mignotte, Zentralblatt MATH, Vol. 1030, 2004)

"This is Barbeau's second book in the Springer Problem Books in Mathematics series ... . Brief hints are given at the end of each chapter and Answers and Solutions are given at the end of the book ... . All of the material in the book is aimed at undergraduate level. ... The book does provide a good source of exercises and interesting ideas for student projects, as well as giving a reasonably thorough account of the solutions, applications and generalisations of Pell's Equation." (Peter G. Brown, The Australian Mathematical Society Gazette, Vol. 30 (4), 2003)

"The present book is described by its author as a 'focussed exercise book in algebra', and is aimed at both college students and talented sixth-formers, with the particular intention of providing training in the key skills of manipulating algebraic expressions judiciously, and with a sense of strategy. ... readers are encouraged to do much of the exploring for themselves through carefully guided examples ... . The book is full of interest, and largely succeeds in its aims." (Michael Ward, The Mathematical Gazette, Vol. 88 (512), 2004)

"Ed Barbeau has prepared a 'focused exercise book in algebra' based on various aspects of Pell's equation. ... The author does a wonderful job in preparing and motivating the reader ... . An array of interesting related problems, facts, and 'explorations' are found here. ... This book is a stepping stone towards any one of the several related areas of study, including diophantine analysis, diophantine approximation, algebraic number theory ... . It is perfectly suited for any first course in number theory." (Gary Walsh, SIAM Review, Vol. 46 (1), 2004)

"The author's book, as the title indicates, is primarily concerned with aspects of Pell's equations ... . The book has nine chapters ... followed by extensive answers, comments and solutions to posed problems and explorations, a glossary of several pages, separate references to books and papers, and a brief index. ... would also be suitable for any person interested in number theory. ... The book reveals the enduring interest by mathematicians in Pell's equation ... ." (George W. Grossman, Mathematical Reviews, 2004f)

"Barbeau develops the theory of Pell's equation (a piece of quadratic form theory) entirely as a series of exercises. ... The book's form recommends it for shepherding undergraduates to research, and is a good source for higher-degree analogs of Pell's equation. Includes problem answers and solutions." (D.V. Feldman, CHOICE, December, 2003)

"Pell's equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. ... The topic is motivated and developed through sections of exercises that allow students to recreate known theory and provide a focus for their algebraic practice. There are also several explorations that encourage readers to embark on their own research." (Zentralblatt für Didaktik der Mathematik, August, 2003)

"Pell's equations are ... as old as mathematics, but the theory of these equations is a modern branch of mathematics development. ... This book is a curious exercise book, but it is much more. ... The exercises, explorations are well-chosen, showing the real nature of mathematical thinking ... . Without any special background you will enjoy the problems throwing you in the very heart of this science. ... Whoever you are ... you will find some beautiful, mind awakening problems in this book." (Lajos Pintér, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

Spis treści

* The square root of 2 * Problems leading to Pell's equation and preliminary investigations * Quadratic surds * The fundamental solution * Tracking down the fundamental solution * Pell's equation and pythagorean triples * The cubic analogue of Pell's equation * Analogues of the fourth and higher degrees * A finite version of Pell's equation * Answers and solutions